Prove: Square Matrix Can Be Written As A Sum Of A Symmetric And Skew-Symmetric Matrices

Solution 1:

If $A$ is a square matrix in $\mathcal M_n(\Bbb C)$ then $$A=\underbrace{\frac12(A+A^t)}_{\text{symmetric matrix}}+\underbrace{\frac12(A-A^t)}_{\text{skew-symmetric matrix}}$$

Solution 2:

$$(A+A^t)_{ij} =A_{ij}+A^t_{ij} = A^t_{ji}+A_{ji} = (A^t+A)_{ji} = (A+A^t)_{ji}$$ Therefore $(A+A^t)$ is symmetric. $$(A-A^t)_{ij} =A_{ij}-A^t_{ij} = A^t_{ji}-A_{ji} = (A^t-A)_{ji} = -(A-A^t)_{ji}$$ Therefore $(A-A^t)$ is scew-symmetric. Since $$A = \frac12A+\frac12A+\frac12A^t-\frac12A^t=\frac12(A+A^t)+\frac12(A-A^t)$$ $A$ can be Written as a sum of a scew-symmetric and a symmetric Matrix.